1 Combinations/Permutations in R
2 Generating Events in R
We can roll a single dice as follows:
We can also roll many dice:
3 Combinations and Permutations in R
How many ways can we choose 2 items from a list of 4 things?
We can look at all of these combinations using the combn() function:
What if order matters? In this case, there are \(2!\) times as many ways to do it (that’s “two factorial”, not just me yelling the word “TWO!!!”, although in this case they’re the same number).
Notice that there are twice as many possibilities! In the output of combn(), we could take each column and just reverse the order. This gives us 6 more possibilities.
If we were choosing three items, then each combination could be rearranged in six different permutations. In this case, the number of permutations is 6 (3!) times as large as the number of combinations.
4 All Combinations of Different Variables (Tree Diagrams)
The expand.grid() function can be very helpful with arbitrary collections of things. For the cars example in class:
5 Probabilities
Probabilities as long-run frequencies: We used some code to generate 100 dice rolls before. We can use this to demonstrate probability!
If we were to roll 6,000 dice, how many of them would show a 1? Note that the following two statements are different!
- A 1 should show up in 1/6 rolls, so we expect approximately 1,000 1s.
- If we get 1,000 1s, then the probability is approximately 1/6.
Our definition is based on the second point, but instead of 6,000 times we roll the dice an infinite number of times (although I don’t recommend literally doing this, that will hurt your wrist).
Probabilities from theory: For the combinations/permutations, we can basically use R as a calculator.
L02 notes Example 1.3.1: What is the probability of drawing an ace from a well-shuffled deck of 52 playing cards?
L02 notes Example 1.3.2: Find the probability that a randomly chosen Lotto Max entry wins the Jackpot.
L02 notes Example 1.3.2: Probability of matching 6 out of 7 numbers: \(P(6\text{ out of }7) = \dfrac{\phantom{1}_{7}C_6 \phantom{1}_{42}C_1}{ \phantom{1}_{49}C_7}\)